Search results for "Complex plane"

Singularity tracking for Camassa-Holm and Prandtl's equations

2006

In this paper we consider the phenomenon of singularity formation for the Camassa-Holm equation and for Prandtl's equations. We solve these equations using spectral methods. Then we track the singularity in the complex plane estimating the rate of decay of the Fourier spectrum. This method allows us to follow the process of the singularity formation as the singularity approaches the real axis.

D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

2015

The D-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group GL(2, C) of invertible 2 × 2 matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions…

Massless positivity in graviton exchange

2021

We formulate Positivity Bounds for scattering amplitudes including exchange of massless particles. We generalize the standard construction through dispersion relations to include the presence of a branch cut along the real axis in the complex plane for the Maldestam variable $s$. In general, validity of these bounds require the cancellation of divergences in the forward limit of the amplitude, proportional to $t^{-1}$ and $\log(t)$. We show that this is possible in the case of gravitons if one assumes a Regge behavior of the amplitude at high energies below the Planck scale, as previously suggested in the literature, and that the concrete UV behaviour of the amplitude is uniquely determined…

Low-temperature spectrum of correlation lengths of the XXZ chain in the antiferromagnetic massive regime

2015

We consider the spectrum of correlation lengths of the spin-$\frac{1}{2}$ XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are determined by integrals over certain auxiliary functions and by their zeros. The auxiliary functions satisfy nonlinear integral equations. We analyse these nonlinear integral equations in the low-temperature limit. In this limit we can determine the auxiliary functions and the expressions for the eigenvalues as functions of a finite number of parameters which satisfy finite sets of algebraic equations, the so-called higher-level Bethe Ansatz equations. The behavio…

Quasiconformally Bi-Homogeneous Compacta in the Complex Plane

1999

Sobolev homeomorphic extensions

2021

Let $\mathbb X$ and $\mathbb Y$ be $\ell$-connected Jordan domains, $\ell \in \mathbb N$, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism $\varphi \colon \partial \mathbb X \to \partial \mathbb Y$ admits a Sobolev homeomorphic extension $h \colon \overline{\mathbb X} \to \overline{\mathbb Y}$ in $W^{1,1} (\mathbb X, \mathbb C)$. If instead $\mathbb X$ has $s$-hyperbolic growth with $s>p-1$, we show the existence of such an extension lies in the Sobolev class $W^{1,p} (\mathbb X, \mathbb C)$ for $p\in (1,2)$. Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of $W^{…

A family of complex potentials with real spectrum

1999

We consider a two-parameter non-Hermitian quantum mechanical Hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations. Numerical investigation shows that for some values of the potential parameters the Hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.

Combinatorics of generalized Bethe equations

2012

A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over \({\mathbb{Z}^M}\), and on the other hand, they count integer points in certain M-dimensional polytopes.

Application of the ADR method for the evaluation of the scattering matrix of an open ended coaxial

2007

The use of ADR method for locating poles near the real axis is proposed when the scattering matrix for an open ended coaxial is analysed radiating over a multilayer structure ended in a perfect electrical wall. So a better and faster integral evaluation is made once the subintegral function behaviour is known.

Local moment problem

2014

The work is devoted to the local moment problem, which consists in finding of non-decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)